Fungrim entry: d35c54

\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x - \varepsilon i\right)\right] = \overline{\log G(x)} = \log G(x) - n \left(n - 1\right) \pi i\; \text{ where } n = \left\lfloor x \right\rfloor

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

TeX:

\lim_{\varepsilon \to {0}^{+}} \left[\log G\!\left(x - \varepsilon i\right)\right] = \overline{\log G(x)} = \log G(x) - n \left(n - 1\right) \pi i\; \text{ where } n = \left\lfloor x \right\rfloor

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x < 0 \;\mathbin{\operatorname{and}}\; x \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`RightLimit`	$\lim_{x \to {a}^{+}} f(x)$	Limiting value, from the right
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`ConstI`	$i$	Imaginary unit
`Conjugate`	$\overline{z}$	Complex conjugate
`Pi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d35c54"),
    Formula(Equal(RightLimit(Brackets(LogBarnesG(Sub(x, Mul(epsilon, ConstI)))), For(epsilon, 0)), Conjugate(LogBarnesG(x)), Where(Sub(LogBarnesG(x), Mul(Mul(Mul(n, Sub(n, 1)), Pi), ConstI)), Equal(n, Floor(x))))),
    Variables(x),
    Assumptions(And(Element(x, RR), Less(x, 0), NotElement(x, ZZ))))

Topics using this entry

Barnes G-function